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Exceptional versus $τ$-exceptional sequences for the Auslander algebra of $K[x]/(x^t)$

Maximilian Kaipel

TL;DR

The paper investigates the Auslander algebra $\mathscr{A}_t$ of $K[x]/(x^t)$ and shows that every complete exceptional sequence in $\mathrm{mod}\,\mathscr{A}_t$ is a complete $\tau$-exceptional sequence, connected via an explicit bijection $\Phi$ between basic tilting modules and complete exceptional sequences. It then develops and compares mutation theories: left and right mutations of complete $\tau$-exceptional sequences are defined and shown to preserve completeness, and, whenever a mutation of a complete exceptional sequence remains in $\mathrm{mod}\,\mathscr{A}_t$, it coincides with the corresponding $\tau$-exceptional mutation. A central result is that $\Phi$ commutes with mutations, so tilting mutations lift to mutations of complete exceptional sequences and to $\tau$-exceptional mutations, aligning the classical tilting picture with the $\tau$-tilting framework for $\mathscr{A}_t$. An explicit $t=3$ example confirms the compatibility and elucidates the interplay among tilting modules, exceptional sequences, and $\tau$-exceptional sequences. These results bridge classical representation theory with $\tau$-tilting theory in a concrete and computable setting, clarifying mutation behavior for Auslander algebras.

Abstract

For $\mathcal{A}_t$, the Auslander algebra of $K[x]/(x^t)$, it is shown that every complete exceptional sequence of $\mathcal{A}_t$-modules is a complete $τ$-exceptional sequence. Moreover, it is established that the mutation of complete $τ$-exceptional sequences generalises the mutation of complete exceptional sequences in the category of $\mathcal{A}_t$-modules.

Exceptional versus $τ$-exceptional sequences for the Auslander algebra of $K[x]/(x^t)$

TL;DR

The paper investigates the Auslander algebra of and shows that every complete exceptional sequence in is a complete -exceptional sequence, connected via an explicit bijection between basic tilting modules and complete exceptional sequences. It then develops and compares mutation theories: left and right mutations of complete -exceptional sequences are defined and shown to preserve completeness, and, whenever a mutation of a complete exceptional sequence remains in , it coincides with the corresponding -exceptional mutation. A central result is that commutes with mutations, so tilting mutations lift to mutations of complete exceptional sequences and to -exceptional mutations, aligning the classical tilting picture with the -tilting framework for . An explicit example confirms the compatibility and elucidates the interplay among tilting modules, exceptional sequences, and -exceptional sequences. These results bridge classical representation theory with -tilting theory in a concrete and computable setting, clarifying mutation behavior for Auslander algebras.

Abstract

For , the Auslander algebra of , it is shown that every complete exceptional sequence of -modules is a complete -exceptional sequence. Moreover, it is established that the mutation of complete -exceptional sequences generalises the mutation of complete exceptional sequences in the category of -modules.
Paper Structure (7 sections, 27 theorems, 65 equations, 4 figures)

This paper contains 7 sections, 27 theorems, 65 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Comparing exceptional and $\tau$-exceptional sequences
  4. Comparing mutations
  5. Mutation of $\tau$-exceptional sequences
  6. Relationship with the mutation of ($\tau$-)tilting modules
  7. Example

Key Result

Theorem 1.1

(thm:excepistauexcep) Every complete exceptional sequence in $\mathop{\mathrm{mod}}\nolimits \mathscr{A}_t$ is a $\tau$-exceptional sequence. More precisely, there is an explicit bijection given by the bijection between TF-ordered $\tau$-tilting modules and complete $\tau$-exceptional sequences of MendozaTreffinger.

Figures (4)

  • Figure 1: Overview of the relationship between tilting modules, exceptional sequences and $\tau$-exceptional sequences in $\mathop{\mathrm{mod}}\nolimits \mathscr{A}_t$ and their mutations.
  • Figure 2: Hasse diagram of support $\tau$-tilting modules (left) with tilting modules highlighted in purple and left mutation graph of complete $\tau$-exceptional sequences (right) with exceptional sequences highlighted in purple.
  • Figure 3: The embedding of tilting $\mathscr{A}_3$-modules (purple) into part of the exchange quiver of $\tau$-tilting pairs in $\mathop{\mathrm{mod}}\nolimits \mathscr{A}_3$. Tilting modules are written in their quasi-hereditary decomposition.
  • Figure 4: A subset of the complete $\tau$-exceptional sequences of $\mathop{\mathrm{mod}}\nolimits \mathscr{A}_3$ with arrows corresponding to left $\varphi$-mutations. The set of complete exceptional sequences and the corresponding $\psi$-mutation in $\mathop{\mathrm{mod}}\nolimits \mathscr{A}_3$ form a subset which is highlighted in purple.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Example 1.4
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 47 more